Wavelet transforms are widely used in analysis, where they are known as “multiresolution analysis”, and in image and audio compression, where they are used as a pyramid coding method for lossy compression. The wavelets used are generally from a very small set of analytically designed wavelets, such as Daubechies wavelets, or quadrature mirror filters (“QMF”). For some applications, designing specific wavelets with special coding properties would be beneficial.
Moments of a random variable provide in formation about the shape of distribution. The n-th moment of the distrubution is th value of the n-th power of the devaiations form a particular fixed value. The first order moment is the average value (mean) for the distrubution of the values of the random variable form zero. The mean is the sum of the distances from zero time the probability of th values being at that distance. The second order moment is spread (variance) of the distrubution of values with respect to the mean. The variance os the sum of the squared (n=2) distances from the mean times the probability of the values being at that distance. The square root of the variance is referred to as the standard deviation. Moments of order greater than two include the asymmetry (skewness) of the distribution around the mean (third order) and the flatness (kurtosis) of the distrubution with respect to the normal distribution (fourth order). In general, the term “higher order moment ” or “higher moment” designates those moments where n is greater than two.
Presently, the recreation of data from higher order moments has been considered. However, the solutions proposed had problems working for other than model problems, and the solutions proposed used a relaxation procedure to try to guess the information for the inversion.